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This study studies the synchronisation of two Boolean networks (BNs) coupled in the drive-response configuration, and presents a number of new results. A new indicator variable, which plays an important role in characterising the synchronised nodes between the drive BN and the response BN, is defined by coordinate equivalence. Consequently, some necessary and sufficient conditions are derived for the general synchronisation of the drive-response BNs. Moreover, the upper bound of time to check the criterion is given. Two examples are provided to illustrate the efficiency of the obtained results.

In this study, the controllability of dynamic-algebraic Boolean control networks is investigated based on a new normalisation approach. Some lower dimensional controllability matrices are defined, and new necessary and sufficient conditions for the controllability are presented as well. An example is given to illustrate the efficiency of the proposed results.

This study addresses the observability of Boolean networks (BNs), using semi-tensor product (STP) of matrices. First, unobservable states can be divided into two types, and the first type of unobservable states can be easily determined by blocking idea. Second, it is found that all states reaching to observable states are observable. Based on subgraph of transition matrix and blocking idea, the second type of unobservable states can be also determined. Approaches are obtained to directly determine some observable or unobservable states. An algorithm is designed for determining the observability of BNs as well. Examples are given to illustrate the effectiveness of the given results.